Properties

Label 2-690-23.9-c1-0-9
Degree $2$
Conductor $690$
Sign $0.906 + 0.421i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.959 − 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (−0.841 − 0.540i)6-s + (0.0700 − 0.153i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (5.28 + 1.55i)11-s + (−0.959 − 0.281i)12-s + (1.37 + 3.01i)13-s + (0.0239 − 0.166i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (−3.71 − 2.38i)17-s + ⋯
L(s)  = 1  + (0.678 − 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (0.0636 + 0.442i)5-s + (−0.343 − 0.220i)6-s + (0.0264 − 0.0579i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.131 + 0.287i)10-s + (1.59 + 0.467i)11-s + (−0.276 − 0.0813i)12-s + (0.381 + 0.835i)13-s + (0.00641 − 0.0445i)14-s + (0.169 − 0.195i)15-s + (0.103 − 0.227i)16-s + (−0.901 − 0.579i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16067 - 0.477264i\)
\(L(\frac12)\) \(\approx\) \(2.16067 - 0.477264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-3.07 + 3.67i)T \)
good7 \( 1 + (-0.0700 + 0.153i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-5.28 - 1.55i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-1.37 - 3.01i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (3.71 + 2.38i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-3.75 + 2.41i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-3.33 - 2.14i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (2.47 - 2.85i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-1.34 + 9.36i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (1.37 + 9.53i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-7.12 - 8.21i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 8.61T + 47T^{2} \)
53 \( 1 + (4.59 - 10.0i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (1.51 + 3.32i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-3.28 + 3.79i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (13.1 - 3.84i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (9.05 - 2.65i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-8.80 + 5.65i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-5.83 - 12.7i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.07 - 7.46i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (0.489 + 0.564i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (1.17 + 8.17i)T + (-93.0 + 27.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89950639545826862277875638156, −9.477395319107332818764615196522, −8.920958209002146798990049285318, −7.27511542757619882299741926435, −6.82559456256049313018769726153, −6.07301465693912489875206918993, −4.80814292563364170671378745634, −3.98339880033579781950538280179, −2.63964894189498144237656238046, −1.35065658307288757789595438859, 1.34077086324487606727505555202, 3.25426513563246879150009286166, 4.07302740662840370958420638902, 5.08173268607138634414328275931, 5.99645431216064349677896081807, 6.64100525702543611741950121346, 7.937231377757989293527475882457, 8.823822024640470302652883086036, 9.658277369147363487158676729599, 10.67567889254225147657279133424

Graph of the $Z$-function along the critical line